# Tag Info

Accepted

• 11.8k
Accepted

### Attaching a copy of $S^1$ at each point of $[0, 1]$ and $S^1$ and the fundamental groups of the resulting spaces

The space $B$ is indeed the wedge sum of uncountably many circles, hence its fundamental group is the free group on uncountably many generators, represented by the wedge summands. The space $C$ is ...
• 11.8k

### Seifert-van Kampen theorem, classical version

It tells you: $j_1:U\subset X$ and $j_2:V\subset X$ induce maps on $\pi_1$, $\pi_1(U)\to\pi_1(X)$ and $\pi_1(V)\to\pi_1(X)$, and these are genuine homomorphisms, and if you have two groups $A,B$ and a ...
• 40.5k

• 26.2k
Accepted

### Filtration of wedge of spheres

Let $C$ be the cone on $\mathbb{R}P^2$; then you should be able to find a filtration of that which yields torsion in homology. Since $C$ is contractible, $C \vee S^3 \vee S^3$ has the homotopy type of ...
• 7,687
Accepted

### Homotopy in the space of linear isometries

Let's take $u \in \Bbb{R}^\infty$ and compute (with respecto to the usual metric / Euclidian inner product / norm) :  \begin{align} \|(t\alpha + (1-t)\operatorname{id}_{\Bbb{R}^\infty})(u)\|^2 &...
• 8,353
1 vote
Accepted

• 21k
1 vote
Accepted

### Equivalence between the category $\text{Ho}(\mathcal{M})$ and $\text{Ho}(\mathcal{M}_{\mathcal{F}})$

I don't think there is a substantial difference in complexity between showing full and faithfulness and showing an inverse functor exists. Note that a morphism $f:X\rightarrow Y$ gives rise to a ...

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